DOCSLIB.ORG

Sophus Lie, the Mathematician

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sophus Lie, the Mathematician Courtesy of Scandinavian University Press. Used with permission. Sophus Lie, the mathematician by Sigurdur Helgason Sophus Lie is known among all mathematicians as the founder of the theory of transformation groups which then gave rise to the mod- ern theory of the so-called Lie groups. This is in fact a subject which permeates many branches of modern mathematics and mathematical physics. However in order to see Lie's work in perspective it is im- portant to realize that his first mathematical love was geometry and that this attachment remained with him all his life. I shall therefore try to describe at least some of his work in chronological order; in this way we can better understand the motivations to various stages in his work and see why his ideas took a long time to be assimilated by the mathematical public, at least longer time than he himself would have liked. -.-- 'Since my remarks are aimed at a general audience I have kept math- ematical technicalities at a minimum. Other lectures at this conference will deal with Lie's work in considerable detail so I shall confine myself to selected highlights. In view of the intensity with which Lie carried on his mathematical investigations in his later career it is remarkable that his calling to research in mathematics did not take place until 1868 (age 26), three years after he had finished his university studies in the natural sciences. Yet once in 1862, Sylow was a substitute teacher at the university and gave a lecture series on Galois Theory. At that time this does not seem to have made a very deep impression on Lie, but later developments suggest that it was well etched in his memory. Sylow relates that in 1868, Lie asked him to lend him his old lecture notes on Galois Theory. Considering the fact that his research period was limited to 30 years., his mathematical production is most impressive; yet many unwritten papers and unworked ideas went "withhim to the grave. Nevertheless, 3 in contrast to say Gauss, he was generally eager to publish his ideas and results quickly. The presentation often suffered because of this (he himself would on occasion admit to "leichtsinniger Darstellung" about some of his papers) and this meant that some of his ideas would first be understood by the mathematical public through works of others. This was often a source of disappointment to him; thus in a letter to .Li'ag-]Leffier 1882 he writes: "It is strange that those of my works, which are without comparison the most original and far-reaching, re­ ceive much less attention than the less important ones", not an unusual phenomenon. During the three years after his university degree he gave private lessons in mathematics and once gave a series of popular lectures on astronomy. At the same time he speculated a good deal about geometry and in 1868 he came across the works of J. Poncelet and J. Plucker. These seem to have inspired Lie in a very decisive fashion. These works were: J. Poncelet: (i) Traite des propriet6s projectives des figures (1822) (ii) Theorie des polaires rciproques (Crelle's Journal 1829). J. Plicker: System der Geometrie des Raumes in neuer analytischer Behandlungsweise (Crelle's Journal 1846). Poncelet's 1822 book originated to some extent in the prison at Sara­ tov near Volga; Poncelet was an officer in Napoleon's Russia campaign and was captured. One of the innovations of Poncelet's was an intro­ duction and use of complex numbers in projective geometry; thereby a nondegenerate conic in homogeneous coordinates A,42 + Bxy + Cy2 + Dxz + Eyz + Fz2 = 0 and a line ax + by + cz, = 0 will always intersect in two points. In Plicker's paper geometric figures are no longer a collection of points but geometry becomes just as much a study of families of lines or of spheres etc. To explain this we consider two points in space with homogeneous coordinates X = (xl, 2, 3, 4) , Y = (Y1,Y, Y3, Y4) and put Pik = Xiyk -yixk . Up to a factor the six Pik are the homogeneous coordinates of the line joining X and Y (independent of the choice of coordinates (xi) and (yj) and they satisfy the identity P = P12P34+ P13P42 + P14P23 = 0 ­ Thus the 4-dimensional set of lines in the three-space is represented as a quadric of signature (3,3) in five-dimensional projective space. Two lines intersect if the corresponding points on the quadric lie on the same line in the quadric. One can now study the geometric meaning of equations like E aikpik = 0 etc. and this represents the rudiments of Plicker's line geometry. Curves, surfaces and three-dimensional bodies in point geometry are replaced by their respective analogs in line ge­ ometry, called line surfaces, line congruences and line complexes. For example (P12, 13, P14, P34, P42, P23) = (t,O 1, t, 0, -t 2 ) is one of the families of generating lines for a hyperboloid. It is a line surface. The set of tangents to a surface is a line complex. Lie retained a life long admiration for Poncelet and Pliicker, whom he never met, they having died in 1867 and 1868, respectively. Never­ theless, Lie considered himself a student of Pliicker's. In 1869 Lie writes his first paper "Representation der Imaginiren der Plangeometrie" Here he considers a mapping T given by T(x + yi, z + pi) = (x, y, z), the '"weight" p being fixed. According to Lie, "Every theorem in plane geometry should thereby be special case of a stereometric double the­ orem in the geometry of line congruences". Lie published the paper privately and sent a copy along with an application to the Collegium Academicum in Christiania for a travelling fellowship. He was also ea­ ger to assure himself of priority for these discoveries. Although few seem to have understood the work (it was really a tersely written research announcement), Lie had already earned a certain reputation and the request for a fellowship was granted, enabling Lie to visit the mathe­ matical capitals of Europe. In Lie's collected works edited by Engel and Heegaard this 8 page paper (together with its expanded form) is given a commentary of over 100 pages. Having received the fellowship Lie spent the winter 1869-1870 in Berlin where he came into close contact with Felix Klein and they be­ came good friends. Klein was very much involved in line geometry at 5 the time, having been a student of Pliicker's. Klein had a great talent for absorbing new ideas and finding interconnection between them; Lie was more immersed in the wealth of his own ideas and perhaps felt the need to catch up in his research. After all, Klein had already obtained his doctor's degree whereas Lie had not, although he was seven years older. In Kummer's seminar, Lie gave several lectures on his work on the Reye complex (the set of lines intersecting the faces of a tetrahedron in a given cross ratio), an outgrowth of his first paper. For example he proved that it is an orbit under a certain group fixing the vertices of the polyhedron. This gave rise to joint work of Klein and Lie on what they called r' . curves (curves whose tangents are contained in the R.eye complex). In the Spring of 1870 they both travelled to Paris; taking adjoining rooms in a hotel. In Paris they got acquainted with the group I theorist Camille Jordan and the differential geometer Gaston Darboux. Jordan had just published his paper "Memoire sur les groupes de mouve­ ments' which undoubtedly provided Klein and Lie with much stimulus for the transformation group I viewpoint in their work. I It was in early July 1870 that Lie made his most famous geometric Ir discovery, namely what is now called Lie's line-sphere transformation. II I would like to comment on this in some detail. I For this Lie defines a "sphere geometry" la Pluicker's line geometry. I Consider a sphere with the equation 2 +y2 +z - 2ax- 2by- 2cz + D = 0. In'addition to a, b, c, D we introduce a fifth quantity, the radius r sat­ isfying r2 = a 2 + b2 +c 2 - D. These quantities are to be considered as the coordinates in the space of spheres. We now introduce the corresponding homogeneous coordinates by a = / brl/v, c =/Lv, r = A/l, D = /, and then we have the connection t = E2 + 772 + 2 _A 2 _ /v = 0. Note that points are included (A = 0) and so are hyperplanes (for v = 0). Recall now the Pliicker coordinates (Pik) in line geometry with the connection P = 0. Given the sphere coordinates above we define ho­ mogeneous line' coordinates (Pik) by p2 = + i77, p 34 = - i77, P13 = A, P 42 = - A, P14 = 1, P23 = -V. Then P = 0 as a consequence of the relation 4( = 0. This is Lie's line-sphere transformation in analytic formulation. It has the crucial property that intersecting lines correspond to spheres that are tangential. In the transformation above the coordinates are necessarily complex. The group SL(4, C) acts via projective transformations on the space of lines in C 3 and SO(6, C) acts on the projective bundle associated with the tangent bundle of C 3. The Lie line-sphere transformation is a realization of the local isomorphism of these two groups. Lie himself drew the following consequence of the transformation for surface theory where one has the concept of asymptotic lines (on which the second fundamental form vanishes) and the lines of curvature (curves whose directions are where the curvature is extremal) (see Lie [22], Vol.
Load more

Recommended publications
  • The History of the Abel Prize and the Honorary Abel Prize the History of the Abel Prize
    The History of the Abel Prize and the Honorary Abel Prize The History of the Abel Prize Arild Stubhaug On the bicentennial of Niels Henrik Abel’s birth in 2002, the Norwegian Govern- ment decided to establish a memorial fund of NOK 200 million. The chief purpose of the fund was to lay the financial groundwork for an annual international prize of NOK 6 million to one or more mathematicians for outstanding scientific work. The prize was awarded for the first time in 2003. That is the history in brief of the Abel Prize as we know it today. Behind this government decision to commemorate and honor the country’s great mathematician, however, lies a more than hundred year old wish and a short and intense period of activity. Volumes of Abel’s collected works were published in 1839 and 1881. The first was edited by Bernt Michael Holmboe (Abel’s teacher), the second by Sophus Lie and Ludvig Sylow. Both editions were paid for with public funds and published to honor the famous scientist. The first time that there was a discussion in a broader context about honoring Niels Henrik Abel’s memory, was at the meeting of Scan- dinavian natural scientists in Norway’s capital in 1886. These meetings of natural scientists, which were held alternately in each of the Scandinavian capitals (with the exception of the very first meeting in 1839, which took place in Gothenburg, Swe- den), were the most important fora for Scandinavian natural scientists. The meeting in 1886 in Oslo (called Christiania at the time) was the 13th in the series.
    [Show full text]
  • Institut F¨Ur Informatik Und Praktische Mathematik Christian-Albrechts-Universit¨At Kiel
    INSTITUT FUR¨ INFORMATIK UND PRAKTISCHE MATHEMATIK Pose Estimation Revisited Bodo Rosenhahn Bericht Nr. 0308 September 2003 CHRISTIAN-ALBRECHTS-UNIVERSITAT¨ KIEL Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universit¨at zu Kiel Olshausenstr. 40 D – 24098 Kiel Pose Estimation Revisited Bodo Rosenhahn Bericht Nr. 0308 September 2003 e-mail: [email protected] “Dieser Bericht gibt den Inhalt der Dissertation wieder, die der Verfasser im April 2003 bei der Technischen Fakult¨at der Christian-Albrechts-Universit¨at zu Kiel eingereicht hat. Datum der Disputation: 19. August 2003.” 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. D. Betten (Kiel) 3. Gutachter Dr. L. Dorst (Amsterdam) Datum der m¨undlichen Pr¨ufung: 19.08.2003 ABSTRACT The presented thesis deals with the 2D-3D pose estimation problem. Pose estimation means to estimate the relative position and orientation of a 3D ob- ject with respect to a reference camera system. The main focus concentrates on the geometric modeling and application of the pose problem. To deal with the different geometric spaces (Euclidean, affine and projective ones), a homogeneous model for conformal geometry is applied in the geometric algebra framework. It allows for a compact and linear modeling of the pose scenario. In the chosen embedding of the pose problem, a rigid body motion is represented as an orthogonal transformation whose parameters can be es- timated efficiently in the corresponding Lie algebra. In addition, the chosen algebraic embedding allows the modeling of extended features derived from sphere concepts in contrast to point concepts used in classical vector cal- culus.
    [Show full text]
  • Norway Establishes Abel Prize in Mathematics, Volume 49, Number 1
    Norway Establishes Abel Prize in Mathematics In August 2001, the prime close contact with its minister of Norway an- counterpart in Sweden, nounced the establish- which administers the ment of the Abel Prize, a Nobel Prizes. In consul- new international prize tation with the Interna- in mathematics. Named tional Mathematical in honor of the Norwe- Union (IMU), the Norwe- gian mathematician Niels gian Academy will appoint Henrik Abel (1802–1829), a prize selection commit- the prize will be awarded for tee of perhaps five members the first time in 2003. The and a larger scientific advi- prize fund will have an initial sory panel of twenty to thirty capital of 200 million Norwegian members. The panel will generate kroner (about US$23 million). The nominations to be forwarded to the amount of the prize will fluctuate ac- selection committee. The committee cording to the yield of the fund but will be and panel will be international and will in- similar to the amount of a Nobel Prize. The pre- clude a substantial number of mathematicians liminary figure for the Abel Prize is 5 million Nor- from outside Norway. wegian kroner (about US$570,000). Various details of the prize have yet to be worked The idea of an international prize in honor of out, such as whether it can be given to more than Abel was first suggested by the Norwegian math- one individual and whether recipients of the Fields ematician Sophus Lie near the end of the nine- Medal, long thought of as the “Nobel of mathe- teenth century.
    [Show full text]
  • Leonard Eugene Dickson
    LEONARD EUGENE DICKSON Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was a towering mathematician and inspiring teacher. He directed fifty-three doctoral dissertations, including those of 15 women. He wrote 270 papers and 18 books. His original research contributions made him a leading figure in the development of modern algebra, with significant contributions to number theory, finite linear groups, finite fields, and linear associative algebras. Under the direction of Eliakim Moore, Dickson received the first doctorate in mathematics awarded by the University of Chicago (1896). Moore considered Dickson to be the most thoroughly prepared mathematics student he ever had. Dickson spent some time in Europe studying with Sophus Lie in Leipzig and Camille Jordan in Paris. Upon his return to the United States, he taught at the University of California in Berkeley. He had an appointment with the University of Texas in 1899, but at the urging of Moore in 1900 he accepted a professorship at the University of Chicago, where he stayed until 1939. The most famous of Dickson’s books are Linear Groups with an Expansion of the Galois Field Theory (1901) and his three-volume History of the Theory of Numbers (1919 – 1923). The former was hailed as a milestone in the development of modern algebra. The latter runs to some 1600 pages, presenting in minute detail, a nearly complete guide to the subject from Pythagoras (or before 500 BCE) to about 1915. It is still considered the standard reference for everything that happened in number theory during that period. It remains the only source where one can find information as to who did what in various number theory topics.
    [Show full text]
  • Introduction to Representation Theory by Pavel Etingof, Oleg Golberg
    Introduction to representation theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina with historical interludes by Slava Gerovitch Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Contents Chapter 1. Introduction 1 Chapter 2. Basic notions of representation theory 5 x2.1. What is representation theory? 5 x2.2. Algebras 8 x2.3. Representations 9 x2.4. Ideals 15 x2.5. Quotients 15 x2.6. Algebras defined by generators and relations 16 x2.7. Examples of algebras 17 x2.8. Quivers 19 x2.9. Lie algebras 22 x2.10. Historical interlude: Sophus Lie's trials and transformations 26 x2.11. Tensor products 30 x2.12. The tensor algebra 35 x2.13. Hilbert's third problem 36 x2.14. Tensor products and duals of representations of Lie algebras 36 x2.15. Representations of sl(2) 37 iii Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms iv Contents x2.16. Problems on Lie algebras 39 Chapter 3. General results of representation theory 41 x3.1. Subrepresentations in semisimple representations 41 x3.2. The density theorem 43 x3.3. Representations of direct sums of matrix algebras 44 x3.4. Filtrations 45 x3.5. Finite dimensional algebras 46 x3.6. Characters of representations 48 x3.7. The Jordan-H¨oldertheorem 50 x3.8. The Krull-Schmidt theorem 51 x3.9.
    [Show full text]
  • Lie Group Analysis Ccllaassssiiccaall Hheerriittaaggee
    LIE GROUP ANALYSIS CCLLAASSSSIICCAALL HHEERRIITTAAGGEE Edited by Nail H. Ibragimov ALGA Publications LIE GROUP ANALYSIS CLASSICAL HERITAGE Edited by Nail H. Ibragimov Translated by Nail H. Ibragimov Elena D. Ishmakova Roza M. Yakushina ALGA Publications Blekinge Institute of Technology Karlskrona, Sweden c 2004 Nail H. Ibragimov e-mail:° [email protected] http://www.bth.se/alga All rights reserved. No part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, scanning or otherwise, without written permission of the editor. The cover illustration: An example of invariant surfaces Sophus Lie, "Lectures on transformation groups", 1893, p. 414 ISBN 91-7295-996-7 Preface Today, acquaintance with modern group analysis and its classical founda- tions becomes an important part of mathematical culture of anyone con- structing and investigating mathematical models. However, many of clas- sical works in Lie group analysis, e.g. important papers of S.Lie and A.V.BÄacklund written in German and some fundamental papers of L.V. Ovsyannikov written in Russian have been not translated into English till now. The present small collection o®ers an English translation of four fun- damental papers by these authors. I have selected here some of my favorite papers containing profound re- sults signi¯cant for modern group analysis. The ¯rst paper imparts not only Lie's interesting view on the development of the general theory of di®eren- tial equations but also contains Lie's theory of group invariant solutions. His second paper is dedicated to group classi¯cation of second-order linear partial di®erential equations in two variables and can serve as a concise prac- tical guide to the group analysis of partial di®erential equations even today.
    [Show full text]
  • Sophus Lie: a Real Giant in Mathematics by Lizhen Ji*
    Sophus Lie: A Real Giant in Mathematics by Lizhen Ji* Abstract. This article presents a brief introduction other. If we treat discrete or finite groups as spe- to the life and work of Sophus Lie, in particular cial (or degenerate, zero-dimensional) Lie groups, his early fruitful interaction and later conflict with then almost every subject in mathematics uses Lie Felix Klein in connection with the Erlangen program, groups. As H. Poincaré told Lie [25] in October 1882, his productive writing collaboration with Friedrich “all of mathematics is a matter of groups.” It is Engel, and the publication and editing of his collected clear that the importance of groups comes from works. their actions. For a list of topics of group actions, see [17]. Lie theory was the creation of Sophus Lie, and Lie Contents is most famous for it. But Lie’s work is broader than this. What else did Lie achieve besides his work in the 1 Introduction ..................... 66 Lie theory? This might not be so well known. The dif- 2 Some General Comments on Lie and His Impact 67 ferential geometer S. S. Chern wrote in 1992 that “Lie 3 A Glimpse of Lie’s Early Academic Life .... 68 was a great mathematician even without Lie groups” 4 A Mature Lie and His Collaboration with Engel 69 [7]. What did and can Chern mean? We will attempt 5 Lie’s Breakdown and a Final Major Result . 71 to give a summary of some major contributions of 6 An Overview of Lie’s Major Works ....... 72 Lie in §6.
    [Show full text]
  • Appendix A: the Present-Day Meaning of Geometry
    Appendix A: The Present-day Meaning of Geometry …… effringere ut arta Rerum Naturae portarum claustra cupiret. De Rerum Natura (I, 70–71) The 19th century is today known as the century of industrial revolution but equally important is the revolution in mathematics and physics, that took place in 18th and 19th centuries. Actually we can say that the industrial revolution arises from the important technical and scientific progresses. As far as the mathematics is concerned a lot of new arguments appeared, and split a matter that, for more than twenty centuries, has been represented by Euclidean geometry also considered, following Plato and Galileo, as the language for the measure and interpretation of the Nature. Practically the differential calculus, complex numbers, analytic, differential and non-Euclidean geometries, the functions of a complex variable, partial differential equations and group theory emerged and were formalized. Now we briefly see that many of these mathematical arguments are derived as an extension or a criticism of Euclidean geometry. In both cases the conclusion is a complete validation of the logical and operative cogency of Euclidean geometry. Finally a reformulation of the finalization of Euclidean geometry has been made, so that also the ‘‘new geometries’’ are, today, considered in an equivalent way since they all fit with the general definition: Geometry is the study of the invariant properties of figures or, in an equivalent way a geometric property is one shared by all congruent figures. There are many ways to state the congruency (equivalence) between figures, here we recall one that allows us the extensions considered in this book.
    [Show full text]
  • Bäcklund Transformations and Exact Time-Discretizations for Gaudin and Related Models
    Universit´adegli Studi Roma Tre Dipartimento di Fisica via Della Vasca Navale 84, 00146 Roma, Italy B¨acklund Transformations and exact time-discretizations for Gaudin and related models Author: Supervisor: Federico Zullo Prof. Orlando Ragnisco PhD THESIS Ai miei genitori Contents 1 Introduction 6 1.1 An overview of the classical treatment of surface transformations. 7 1.2 TheClairinmethod............................. 15 1.3 The Renaissance of B¨acklund transformations . 18 1.4 B¨acklund transformations and the Lax formalism . 20 1.5 B¨acklund transformations and integrable discretizations......... 23 1.5.1 Integrable discretizations . 23 1.5.2 The approach `ala B¨acklund . 25 1.6 OutlineoftheThesis ............................ 36 2 The Gaudin models 39 2.1 Ashortoverviewonthepairingmodel . 39 2.2 TheGaudingeneralization . 41 2.3 Lax and r-matrixstructures.. 42 2.4 In¨o¨nu-Wigner contraction and poles coalescence on Gaudin models . 47 3 B¨acklund transformations on Gaudin models 51 3.1 Therationalcase .............................. 51 3.1.1 The dressing matrix and the explicit transformations . 53 3.1.2 The generating function of the canonical transformations . 55 3.1.3 Thetwopointsmap ........................ 56 3.1.4 Physical B¨acklund transformations . 60 3.1.5 Interpolating Hamiltonian flow . 61 3.2 Thetrigonometriccase ........................... 63 3.2.1 The dressing matrix and the explicit transformations . 65 3.2.2 Canonicity.............................. 71 5 CONTENTS 3.2.3 Physical B¨acklund transformations . 73 3.2.4 Interpolating Hamiltonian flow . 75 3.2.5 Numerics .............................. 76 3.3 Theellipticcase............................... 79 3.3.1 The dressing matrix and the explicit transformations . 81 3.3.2 Canonicity.............................. 84 3.3.3 Physical B¨acklund transformations .
    [Show full text]
  • Historical Review of Lie Theory 1. the Theory of Lie
    Historical review of Lie Theory 1. The theory of Lie groups and their representations is a vast subject (Bourbaki [Bou] has so far written 9 chapters and 1,200 pages) with an extraordinary range of applications. Some of the greatest mathematicians and physicists of our times have created the tools of the subject that we all use. In this review I shall discuss briefly the modern development of the subject from its historical beginnings in the mid nineteenth century. The origins of Lie theory are geometric and stem from the view of Felix Klein (1849– 1925) that geometry of space is determined by the group of its symmetries. As the notion of space and its geometry evolved from Euclid, Riemann, and Grothendieck to the su- persymmetric world of the physicists, the notions of Lie groups and their representations also expanded correspondingly. The most interesting groups are the semi simple ones, and for them the questions have remained the same throughout this long evolution: what is their structure, where do they act, and what are their representations? 2. The algebraic story: simple Lie algebras and their representations. It was Sopus Lie (1842–1899) who started investigating all possible (local)group actions on manifolds. Lie’s seminal idea was to look at the action infinitesimally. If the local action is by R, it gives rise to a vector field on the manifold which integrates to capture the action of the local group. In the general case we get a Lie algebra of vector fields, which enables us to reconstruct the local group action.
    [Show full text]
  • Acceptance Speech (Pdf)
    Your Majesty, Your Excellencies, ladies and gentlemen: I’m very happy to be in this beautiful country again, and very much honored by this award. The field of mathematics is a marvelous mosaic built up out of contributions by people from many different cultures, speaking many different languages, and stretching back over many hundreds of years. From the beginning mathematics has had a dual nature, partly abstract and self contained, but intimately concerned with understanding of the physical world around us. Much important mathematics was first inspired by real world problems; and mathematics has often contributed in totally unexpected ways. No one could have guessed that Riemann’s study of curvature would form the basis for Einstein’s theory of gravity; or that Hilbert’s theory of infinite dimensional vector spaces would provide the foundations for quantum mechanics. The British mathematician G. H. Hardy proudly bragged that his work in number theory would never be sullied by applications. He would have been horrified to learn that it is now the basis for methods in cryptography which are fundamentally important in commercial applications and also in military applications. The connections between mathematics and other sciences work in both directions. Claude Shannon’s work on communication theory was inspired by the work of physicists on statistical mechanics, and now has important applications not only in computer science but also in the mathematical theory of dynamical systems. The mathematical theory of Riemannn surfaces is now very important to mathematical physicists. Conversely, tools developed by mathematical physicists play a very important role in topology.
    [Show full text]
  • Lie Symmetries of Differential Equations: Classical Results and Recent Contributions
    Symmetry 2010, 2, 658-706; doi:10.3390/sym2020658 OPEN ACCESS symmetry ISSN 2073-8994 www.mdpi.com/journal/symmetry Review Lie Symmetries of Differential Equations: Classical Results and Recent Contributions Francesco Oliveri Department of Mathematics, University of Messina, Contrada Papardo, Viale Ferdinando Stagno d’Alcontres 31, I–98166 Messina, Italy; Tel.: +39-090-6765065; Fax: +39-090-393502; E-Mail: [email protected] Received: 2 January 2010 / Accepted: 30 March 2010 / Published: 8 April 2010 Abstract: Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems, to the derivation of conservation laws, to the construction of links between different differential equations that turn out to be equivalent. This paper reviews some well known results of Lie group analysis, as well as some recent contributions concerned with the transformation of differential equations to equivalent forms useful to investigate applied problems. Keywords: lie point symmetries; invariance of differential equations; invertible mappings between differential equations Classification: MSC 22E70, 34A05, 35A30, 58J70, 58J72. 1. Introduction Symmetry (joined to simplicity) has been, is, and probably will continue to be, an elegant and useful tool in the formulation and exploitation of the laws of nature. The request of symmetry accounts for the regularities of the laws that are independent of some inessential circumstances. For instance, the reproducibility of experiments in different places at different times relies on the invariance of the laws Symmetry 2010, 2 659 of nature under space translation and rotation (homogeneity and isotropy of space), and time translation (homogeneity of time).
    [Show full text]